Coarse-Grained Theory of Surface Nanostructure Formation

Dimitri D. Vvedensky

The Blackett Laboratory, Imperial College London

Christoph A. Haselwandter

Department of Applied Physics, Caltech

Imperial CollegeLondon

Outline

• Systems with many scales• The coarse-graining road map• Renormalization-group trajectories• Transient effects and crossover• Experimental realizations• Extension to the submonolayer regime

Outline

• Systems with many scales• The coarse-graining road map• Renormalization-group trajectories• Transient effects and crossover• Experimental realizations• Extension to the submonolayer regime

Multiscale Physics: Turbulence

C. H. K. Williamson, Cornell (Source: http://www.efluids.com)

• Free gliding of delta-wing in water• Fluorescent dye illuminated by laser• Vortices in near field• Turbulence in far field• Both panels have same scale• Energy cascades from large to small scales

Multiscale Physics: Fracture

F. F. Abraham, D. Brodbeck, R. A. Rafey, andW. E. Rudge, Phys. Rev. Lett. 73, 272 (1994).

• 2D simulation of 106- atom system• Bond-breaking at crack tip• Dislocation emission blunts crack tip• Feed-back between atomic and continuum modes

Multiscale Physics: Critical Phenomena2D Ising Model

J. D. Noh, Chungham National University, Korea

Multiscale “Physics”: Medical ImplantsB. Kasemo, Surface Science 500, 656 (2002)

Time Scale

ns

µm

ms

Outline

• Systems with many scales• The coarse-graining road map• Renormalization-group trajectories• Transient effects and crossover• Experimental realizations• Extension to the submonolayer regime

formulationMaster & Chapman-

Kolmogorov equations

Lattice Langevin equation

Hierarchy of equations

KMC simulations

Lattice model

Macroscopic equation

continuum variables

renormalization-group(crossover, scaling,self-organization)

analytic

stable

fixed point

Chua et al., PRE 72, 051103 (2005),C. A. H. & D. D. V., PRE 76, 041115 (2007)

Direct analysis/solution

C. A. H. & D. D. V. PRL, EPL, PRE (2007, 2008)

Coarse-Graining Road Map

Continuum equation

Lattice-to-Continuum Method

“Atomistic” Continuum Equation

Compare atomistic equation directly to computer simulations?

Extract qualitative multiscale surface features via RG analysis…

Continuum Equation for Random Deposition/Diffusion

The Multiscale Paradigm

M. Scheffler, FHI–Berlin

Outline

• The coarse-graining road map

• Renormalization-group trajectories

• Transient effects and crossover

• Experimental realizations

• Extension to the submonolayer regime

Renormalization Group Transformations

Real Space

MomentumSpace

Original System Coarse Graining Rescaling

Renormalization-Group Equations

• Points along RG trajectory constitute a hierarchy of equations.

• RG “weeds out” terms that become irrelevant as the scale is increased, and absorbs their contributions into other terms.

Outline

• The coarse-graining road map

• Renormalization-group trajectories

• Transient effects and crossover

• Experimental realizations

• Extension to the submonolayer regime

Stable & Unstable Fixed Points

Initial Conditions & Crossover

Outline

• The coarse-graining road map

• Renormalization-group trajectories

• Transient effects and crossover

• Experimental realizations

• Extension to the submonolayer regime

Regimes of Growth

• D/F>>1. Typical MBE conditions.Initially, conserved Mullins-Herring.Submonolayer regime.

• D/F ≈ 1. Diffusion noise diminished in importance. Initially, Mullins-Herring.Al on silicone oil (Fang et al., Thin SolidFilms 517, 3408 (2009)).

• D/F<<1. Growth dominated by shot noise.

Growth on Patterned Substrates. 1.H.-C. Kan et al., Phys. Rev. Lett. 92, 146101 (2004).

KPZ

cVLDS

VLDS

Mound

Growth on Patterned Substrates. 2.H.-C. Kan et al., Phys. Rev. Lett. 92, 146101 (2004).

Experiment KPZ cVLDS

Analysis from Initial Conditions

Outline

• The coarse-graining road map

• Renormalization-group trajectories

• Transient effects and crossover

• Experimental realizations

• Extension to the submonolayer regime

The Submonolayer Regime

Summary

• Continuum formulation that retains connection to atomistic processes

• Unifies a wide range of experimental scenarios

• Extension to submonolayer regime

• Large-scale morphologies on patterned substrates

Ongoing and Future Work

• Coarse-graining magnetohydrodynamics• Coarse-grained molecular dynamics